How do I flip an armature to face the other way?
Make sure you're in object mode. Then rotate the armature instead of the mesh. The mesh will rotate with the armature if you have it parented (and if it isn't, then the armature won't do you any good because it won't deform the mesh!). Then apply the rotation both on the armature and on the mesh using Ctrl-A.

To make your armature deform the mesh, first select your mesh, then your armature and hit Ctrl-P. Choosing "With automatic weights" from the resulting popup is a good first step.